Optimal. Leaf size=94 \[ \frac {x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (b \sinh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac {1}{2},\frac {p+1}{2};\frac {p+3}{2};-\sinh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt {\cosh ^2\left (c+d x^n\right )}} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5322, 5320, 2643} \[ \frac {x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (b \sinh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac {1}{2},\frac {p+1}{2};\frac {p+3}{2};-\sinh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt {\cosh ^2\left (c+d x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 5320
Rule 5322
Rubi steps
\begin {align*} \int (e x)^{-1+n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int (b \sinh (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=\frac {x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \, _2F_1\left (\frac {1}{2},\frac {1+p}{2};\frac {3+p}{2};-\sinh ^2\left (c+d x^n\right )\right ) \left (b \sinh \left (c+d x^n\right )\right )^{1+p}}{b d e n (1+p) \sqrt {\cosh ^2\left (c+d x^n\right )}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 93, normalized size = 0.99 \[ -\frac {x^{-n} (e x)^n \sinh \left (2 \left (c+d x^n\right )\right ) \left (-\sinh ^2\left (c+d x^n\right )\right )^{\frac {1}{2} (-p-1)} \left (b \sinh \left (c+d x^n\right )\right )^p \, _2F_1\left (\frac {1}{2},\frac {1-p}{2};\frac {3}{2};\cosh ^2\left (d x^n+c\right )\right )}{2 d e n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+n} \left (b \sinh \left (c +d \,x^{n}\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\mathrm {sinh}\left (c+d\,x^n\right )\right )}^p\,{\left (e\,x\right )}^{n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sinh {\left (c + d x^{n} \right )}\right )^{p} \left (e x\right )^{n - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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